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Complex Number

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A number of the form $x + iy$ where x and y are real numbers and i$^{2}$ = -1 or i = $\sqrt{-1}$ is called a complex number. $x$ is called the real part of the complex number and y is called its imaginary part. The set of all complex numbers is denoted by $C$ = {$x + iy$ |x, y $\epsilon $ $R$}.
When $y$ is zero, the complex number $x + iy$ reduces to a real number x and when x is zero it reduces iy which is purely an imaginary number. Thus the set of real numbers and set of imaginary numbers are subsets of the set of complex numbers.
Examples: 2 + 5$i$,  075 + 0.95$i$, 1 - 4$i$ are all complex numbers.
In complex numbers definition, the complex numbers are being represented in their own plane called complex plane and the total idea of the two dimension complex plane has been extended from the one dimension number line where real part is represented on the horizontal axis and imaginary part on the vertical axis. There are many applications of complex numbers such as scientific fields, applied mathematics, quantum physics and engineering electromagnetism.

Complex Plane

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A complex number $(x + iy)$ is essentially an ordered pair $(x, y)$ of real numbers. An ordered pair $(x, y)$ of real numbers geometrically represents a point in the $xy$ - plane. Thus there is a one-to-one correspondence between the set of all complex numbers $C$ and the set of all points in the $xy$ plane. Hence the $xy$ - plane whose points $(x, y)$ represent complex numbers of the form $x + iy$ is called the complex plane or Argands plane .
Any point on the x-axis has coordinates of the form $(x, 0)$ and hence represents the complex number $x + i0$ which is a purely real number. Hence the x - axis is referred to as the real axis of the complex plane. Any point on the y axis has coordinates of the form $(0, y)$ and represents a complex number of the form $0 + iy$ which is a purely imaginary number.
Therefore, the y- axis is referred to as the imaginary axis. The origin has coordinates (0, 0) and hence it represents the complex number 0 + i 0.
The diagram in which the complex numbers are represented by points of the complex plane is called Argand's diagram. The point
$(x, y)$ represents $Z$ = $x + iy$ and its image in the real axis is $(x, -y)$ and hence represents the complex number $x - iy$ = $\bar{Z}$.

Complex Conjugate

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If $Z$ = $x + iy$ is a complex number, then $x - iy$ is called the conjugate of $Z$ and is denoted by $\bar{Z}$, and Z and $\bar{Z}$ are called conjugate complex numbers.
The following results follow if $Z$ and $\bar{Z}$ are conjugate complex numbers:
1) $Z$ + $\bar{Z}$ = $(x + iy)$ + $(x - iy)$ = 2$x$

2) $x$ = Real part of $Z$ is $\frac{Z + \bar{Z}}{2}$

3) $y$ = Imaginary part of $Z$ is $\frac{Z - \bar{Z}}{2i}$

4) $Z$$\bar{Z}$ = $(x + iy)$ - $(x - iy)$ = x$^{2}$ + y$^{2}$ = |z|$^{2}$

Operations with Complex Numbers

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Operations on complex numbers are explained below:

Addition of two complex numbers:
If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ are two complex numbers, then their sum denoted by $Z_{1} + Z_{2}$ is defined to be the complex number ($x_{1} + x_{2}$) + i (y$_{1} + y_{2}$)

Multiplication of complex numbers:
If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ then their product denoted by $Z_{1}Z_{2}$ is defined to the complex number $Z_{1}Z_{2}$ = ($x_{1}x_{2} - y_{1}y_{2}) + i(x_{1}y_{2}+x_{2}y_{1}$)

Division of a complex number by a non zero complex number:
If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ $\neq$ 0 then

$\frac{Z_{1}}{Z_{2}}$ = $\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}$

= $\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}$ $\times$ $\frac{x_{2}-iy_{2}}{x_{2}-iy_{2}}$

= $\frac{(x_{1}x_{2}+y_{1}y_{2})+i(y_{1}x_{2}-y_{2}x_{1})}{x_{2}^{2}+y_{2}^{2}}$

which is a complex number

Modulus of a complex number:
If $Z$ = $x + iy$ is a complex number, then the modulus of $Z$ denoted by $|Z|$ is the real number $\sqrt{x^{2}+y^{2}}$

$|Z|$ = $|x + iy|$ = $\sqrt{x^{2}+y^{2}}$

$|Z|$ = $|x - iy|$ = $\sqrt{x^{2}+y^{2}}$ = $| x - iy|$ = $| -Z|$

$|Z|$ = $|-Z|$

Simplifying Complex Numbers

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Example 1: Express $\frac{1}{4 + 3i}$ in the form $x + iy$
Solution :

$\frac{1}{4 + 3i}$ = $\frac{1}{4+3i}$ $\times$ $\frac{4-3i}{4-3i}$

= $\frac{4 - 3i}{16 + 9}$

= $\frac{4}{25}$ + ($\frac{-3}{25}$)$i$

= $\frac{4}{25}$ - $i$ ($\frac{3}{25}$)


Example 2: Find the modulus of $\frac{1}{(2+i)^{2}}$ - $\frac{1}{(2-i)^{2}}$

Solution:

$\frac{1}{(2+i)^{2}}$ - $\frac{1}{(2-i)^{2}}$ = $\frac{1}{3+4i}$ - $\frac{1}{3-4i}$

= $\frac{3-4i}{(3+4i)(3-4i)}$ - $\frac{3+4i}{(3-4i)(3+4i)}$

= $\frac{3-4i}{9+16}$ - $\frac{3+4i}{9+16}$

= $\frac{3-4i-3-4i}{25}$

=$\frac{-8i}{25}$

= 0 - $\frac{8}{25}$

Its modulus = $\sqrt{0+\frac{64}{625}}$ = $\frac{8}{25}$